Problem 25 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Use Venn diagrams to represent the following relationships: a. \(A \subset B\) and \(B \subset C\) b. \(A \subset U\) and \(B \subset U\), where \(A\) and \(B\) have no elements in common c. The sets \(A, B\), and \(C\) are equal.

Step-by-Step Solution

Answer

a. Draw three circles with \(A\) inside \(B\) and \(B\) inside \(C\), indicating \(A \subset B\) and \(B \subset C\). b. Draw a rectangle for the universal set \(U\) and two non-overlapping circles inside for \(A\) and \(B\), indicating \(A \subset U, B \subset U\), and \(A \cap B = \emptyset\). c. Draw three overlapping circles for \(A, B\), and \(C\) such that they completely cover one another, indicating the sets are equal.

Step 1: a. \(A \subset B\) and \(B \subset C\)

First, draw a Venn diagram with three circles, one for each set, \(A\), \(B\), and \(C\). Make sure that circle \(A\) is completely inside circle \(B\), and circle \(B\) is completely inside circle \(C\). This way, the diagram will show \(A \subset B\) and \(B \subset C\).

Step 2: b. \(A \subset U\) and \(B \subset U\), where \(A\) and \(B\) have no elements in common

In this case, draw a rectangle to represent the universal set \(U\). Then, draw two circles inside the rectangle, one for each set, \(A\) and \(B\). Make sure that circles \(A\) and \(B\) do not intersect, meaning they have no elements in common. This way, the diagram shows \(A \subset U\) and \(B \subset U\), with \(A \cap B = \emptyset\).

Step 3: c. The sets \(A, B\), and \(C\) are equal

To represent this relationship, first draw a Venn diagram with three circles, one for each set, \(A\), \(B\), and \(C\). Make sure that all three circles overlap completely, meaning that the area of one circle is covered by the others. This way, the diagram shows that the sets \(A\), \(B\), and \(C\) have the same elements, making them equal.

Key Concepts

Set Theory

Set theory is a fundamental part of mathematics that explores the idea of collections of objects, known as sets. Within set theory, we can define and understand how different sets relate to one another. Sets are typically represented using curly braces, such as \( A = \{1, 2, 3\} \).

The relationships between sets can be depicted visually using Venn diagrams, which show how different sets intersect and relate in a universal space. For example:

• Two overlapping circles can illustrate the common elements shared between two sets.
• Three circles can show relationships among three sets.

These diagrams help simplify complex relationships by providing a visual representation.

Universal Set

In set theory, a universal set \( U \) is the set that contains all possible elements within a particular context or problem. It acts as the 'universe' for all the sets we are considering, providing a complete space from which subsets can be formed.

Venn diagrams often represent the universal set as a rectangle, with all other sets shown as circles within it.

• If \( A \) and \( B \) are subsets of \( U \), then they are completely contained within the universal set.
• Sets are drawn inside the rectangle to show they are part of \( U \).

This concept helps illustrate the idea that no set can exist in isolation without reference to the universal set.

Subset

A subset is a set where all its elements are also contained in another set. If \( A \) is a subset of \( B \), denoted as \( A \subset B \), then every element of \( A \) is also an element of \( B \).

This relationship can be clearly demonstrated using a Venn diagram. Let's look at how they are applied:

• If \( A \subset B \), the circle representing \( A \) should be entirely within the circle representing \( B \).
• Similarly, if \( B \subset C \), the circle for \( B \) should be within that of \( C \).

Subsets are crucial for understanding how different sets integrate within each other.

Equal Sets

Equal sets are sets that contain the exact same elements. If \( A \), \( B \), and \( C \) are equal sets, it implies that they have identical members. All elements of \( A \) are in \( B \), and vice versa, and the same applies to \( C \).

In a Venn diagram, equal sets can be shown by overlapping circles that perfectly coincide, meaning each circle entirely covers the others with no outside areas. This means:

• The area of one circle is fully covered by the other sets.
• Each set contains all the same items, leaving no remainders.

Equal sets illustrate the idea of completeness and sameness among different sets.